Upon taking the mission, enter the marked abandoned house, take out the zombie Walkers and Thug for the "backdoor keys to the weapon store". Be sure to take the Shock Shotgun mod from the shelf before re-entering the gun store.

Inside the gun store, defeat the zombies at close quarters. When you unbar the door to let Jared and Thomas in, the store alarm will sound off and two waves of zombies (Walkers, then Infected), will attack. Once you neutralize all marked enemies, the gun locker will be accessible.

Until you complete more story missions, the gun store in Moresby will spawn random firearms and Military Knives each time you load/cycle the map.